Why don’t any engines use turboelectric oxidiser pumps?

Question

Most larger liquid-fuelled rocket engines¹ use fuel and oxidiser turbopumps driven by the hot, high-pressure gasses produced by burning said fuel and oxidiser:

• The gas-generator cycle burns some of the fuel and oxidiser in its namesake gas generator to produce hot, high-pressure gasses; these gasses are then used to drive the engine’s turbopumps, before being dumped overboard.
• The staged-combustion cycle is similar to the gas-generator cycle, but the gasses from the gas generator (now frequently known instead as a preburner), after being used to drive the turbopumps, are fed back into the main combustion chamber rather than being dumped overboard.
• The combustion-chamber tapoff cycle drives the turbopumps with gasses from the main combustion chamber, rather than using a separate gas generator/preburner.

The fuel turbopumps are fairly easy to deal with; they are almost² always driven by fuel-rich combustion gasses, which are friendly to engine plumbing and turbomachinery and don’t pose a risk of reacting with the fuel if they leak past the pump seals and come into contact with said fuel.

The oxidiser turbopumps, however, are a very different beast. When producing hot, high-pressure combustion gasses to drive an oxidiser turbopump, there are basically three options, each with its own problems:

1. The gas generator can run at the stoichiometric mixture ratio (just enough oxidiser to completely burn the fuel provided, with no extra of either reactant), which has a habit of melting the pump’s turbine wheels.
2. The gas generator can run rich (more fuel and less oxidiser than stoichiometric), which requires really good sealing to prevent the combustion gasses from leaking past the pump seals, coming into contact with the oxidiser, and exploding (or vice versa); this generally necessitates the use of two sets of seals, with the space between the two filled with a nonreactive gas³ at positive pressure relative to both the combustion gasses and the oxidiser.
3. The gas generator can run lean (less fuel and more oxidiser than stoichiometric), which produces large amounts of superheated oxidiser-rich gasses, which are hideously-difficult to deal with, due to their tendency to eat engine plumbing and turbomachinery.

Physically separating the oxidiser pump from the turbomachinery powering it would allow the turbopump to be driven by docile fuel-rich gasses without needing a complex gas-purged seal system; one way of doing this would be to use a turboelectric-drive system, with the combustion-gas-driven turbine driving an electrical generator and the resulting electricity being used to power an electric motor driving the oxidiser turbopump. The electrical transmission between the turbine and the pump would add some mass and slightly reduce the pump’s efficiency (though not by much - well-designed electric motors and generators can have conversion efficiencies well north of 90%), but would eliminate the need for a complicated and heavy gas-purge system or for difficult-to-engineer superheated-oxidiser-containment-and-transport equipment.

Why do no liquid-fuelled rocket engines, to the best of my knowledge, use turboelectric-drive oxidiser turbopumps?

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¹: Smaller liquid-fuelled engines tend to use the pressure-fed or expander cycles, which are very simple but scale poorly, while the newer electric-pump-fed cycle requires heavy battery banks to drive its turbopumps.

²: A very few engines use oxidiser-rich combustion gasses to drive their fuel turbopumps; these are almost exclusively those that already use oxidiser-rich gasses for the oxidiser turbopumps. This arrangement is extremely uncommon, as it combines the disadvantages of oxidiser-turbopump drive methods 2 and 3 without the benefits of either.

³: Generally helium, which is extremely light and almost completely inert (although also extremely expensive).

Answer 1

Working off the Falcon pump power of 7500 kW and power to weight of 10 kW/kg (electric aircraft engine) and assuming generator weighs the same as the motor, that gives a best case motor/generator set for a merlin engine of 1500kg, on a existing engine mass of 750kg so certainly not lighter than the seals and associated plumbing.

Some possible cases where it might make sense is beyond earth Orbit, where things like the ability to start the engine on electric power, fine speed control, possible cross connect for redundancy and ability to just run the generator for electricity might conceivably make up for that mass penalty but these start getting complicated to calculate.